chapter 3 design for manufacture of fabry-perot cavity design for manufacture of fabry-perot cavity

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  • 19

    Chapter 3

    Design for manufacture of Fabry-Perot cavity sensors

    When Fabry-Perot cavity sensors are manufactured, the thickness of each

    layer must be tightly controlled to achieve the target performance of a sensor.

    However, there are unavoidable errors in thickness even though techniques of

    thickness control for thin films have rapidly improved [1] . For Fabry-Perot

    optical interference filters it has long been recognized that the performance of the

    filter is greatly influenced by random thickness variations in the films. For

    instance, the resonant wavelength is very sensitive to thickness variations.

    Degradation in the performance of the filters has been simulated by Monte Carlo

    technique or analytic methods [2, 3] .

    In this chapter the impact of manufacturing-induced variations on the

    performance of Fabry-Perot cavity sensors is studied. In particular, we consider

    how random variations in thickness of the cavity mirrors influence the accuracy

    with which gap can be measured. Through the study, it is found that there exists

    an optimum design of a Fabry-Perot cavity sensor, which is a combination of

    initial cavity gap and mechanical travel of the moving mirror. The optimum

    design gives large manufacturing tolerance to the thickness variations of layers

    and it leads to high accuracy.

  • 20

    3.1 IMPACT OF THICKNESS VARIATIONS ON OPTICAL RESPONSE

    In this section, the variation of the optical response, e.g., of reflectance,

    due to the thickness variation of the layers is calculated using an analytic method.

    This calculation will be a basis for designing Fabry-Perot sensors for

    manufacture, which is to be discussed in the following section. The first

    derivative of the reflectance ( R ) with respect to the thickness of the ith layer can

    be calculated by using the first derivative of the equivalent characteristic matrix

    ( M ). From equation (2.4)

    ∂R ∂zi

    = ∂ ∂zi

    P  

     

    ⋅ P* + P ⋅ ∂ ∂zi

    P*  

     

    , (3.1)

    where P = η in B − C η in B + C

    and P* is the conjugate of P . The partial derivatives of

    B and C are calculated from

    ∂ ∂zi

    B

    C  

     

    = ∂ ∂zi

    M ⋅ 1

    η f 

     

     . (3.2)

    The first derivative of the equivalent characteristic matrix is obtained using

    equation (2.3)

    ∂M ∂zi

    = Mq ⋅ Mq−1 ⋅ ⋅ ∂Mi ∂zi

    ⋅ ⋅M1, (3.3)

    where ∂Mi ∂zi

    = ko ni −sin ko ni zi( ) jηi cos ko ni zi( )

    j ηi cos ko ni zi( ) −sin ko ni zi( )

      

      

    .

  • 21

    Using equations (3.1), (3.2) and (3.3), the first derivatives of the reflectance of a

    Fabry-Perot cavity with respect to each layer can be calculated.

    For the Fabry-Perot cavity sensors described in chapter 2, the first

    derivatives of the reflectances with respect to each layer are calculated as a

    function of gap. Figure 3.1 shows a plot of the first derivative of the reflectance

    of the Fabry-Perot cavity with metal (Au) mirrors. The mirrors of the cavity are

    numbered in ascending order from the fiber, for convenience. From the figure, it

    is obvious that the reflectance of the cavity is the most sensitive to the thickness

    variation of the first Au layer which the incident light meets. Thus, thickness

    control of the first Au layer would be very critical to achieve a target reflectance

    of the cavity.

    As shown in equation (3.3), the first derivative of reflectance with respect

    to a layer also depends on the refractive index of the layer. It implies that for a

    given thickness variation the reflectance curve would be more perturbed by

    thickness variation of a layer with larger refractive index. Figure 3.2 plots the

    first derivatives of the reflectance of a Fabry-Perot cavity with dielectric mirrors

    as described in chapter 2. Each dielectric layer is numbered in ascending order

    from the fiber. For example, nitride 2 represents the second silicon nitride layer

    which the incident light meets. Comparing Figure 3.1 with Figure 3.2, it should

    be noted that the magnitudes of the derivatives with respect to the dielectric

    layers, i.e., silicon dioxide and silicon nitride, are much smaller than ones with

    respect to the metal layers (Au). This indicates that a film with small refractive

    index would be preferred as a mirror layer if manufacturing process for the film

  • 22

    produces the same process-induced thickness variation as a film with large

    refractive index, assuming the layers were of equal thickness.

    Figure 3.1 : The first derivatives of the reflectance of a Fabry-Perot cavity with metal (Au) mirrors, as shown in Figure 2.5. The vertical axis represents the first derivative of the reflectance of the cavity with respect to the thickness of each layer.

  • 23

    Figure 3.2 : The first derivatives of the reflectance of a Fabry-Perot cavity with dielectric films, as shown in Figure 2.6. For simplicity, only the four layers with the largest derivatives are shown.

  • 24

    3.2 DESIGN METHODOLOGY AND CASE STUDY FOR MANUFACTURE

    If Fabry-Perot sensors are to be manufactured in large volume at low cost,

    it will not be practical to individually calibrate every sensor to eliminate

    fabrication-induced response variations. Instead of calibrating every sensor, it

    would be preferable if a nominal response curve could be given for a set of the

    manufactured sensors with some specified accuracy. The accuracy is limited by

    the amount of change in the response curve due to the thickness variation of the

    layers.

    Quantitative treatment of the accuracy of a design is discussed in the

    following subsections. Using the first derivatives obtained in the last section,

    uncertainty in gap induced by thickness variation can be calculated and included

    in the calculation of accuracy. Finally, the accuracy can be used as a metric to

    find an optimum design for manufacture of Fabry-Perot cavity sensors.

    3.2.1 Design Methodology

    For Fabry-Perot sensors, the cavity gap g could be inferred from some

    measurement of reflected light intensity I, that is a function of the reflectance R,

    that is in turn a function of wavelength and the structure of the sensor. For

    example, absolute reflectance of the cavity could be used as a measurand as done

    in chapter 2. The detection scheme is, however, very susceptible to any light

    intensity change in the fiber due to coupling and fiber bending. As an alternative,

    a ratio of the reflectances at two wavelengths (λ1 and λ2) could be used as a

    measurand. Such a detection scheme will be called the dual wavelength

  • 25

    technique in this chapter. This method eliminates errors resulting from

    wavelength-independent changes in the fiber interconnect to the sensor, such as

    fiber bending loss and coupling loss.

    Taking manufacturing-induced variations of layer thickness into account,

    it should be realized that a given measurand value I R,λ( ) could be produced

    from a Fabry-Perot cavity which has different layer thicknesses from a target

    design. This leads to an error in estimating the cavity gap g. In other words, there

    exists an uncertainty ∆g in the gap at a given measurand I R,λ( ) since for mirror layers with thicknesses z1 + ∆z1, z2 + ∆z2 ,⋅ ⋅⋅, zq + ∆zq( ) there can exist a gap thickness g + ∆g that would produce an identical value of I

    I R z1, z2 ,⋅ ⋅⋅,g,⋅ ⋅⋅, zq , λ( )[ ] = I R z1 + ∆z1, z2 + ∆z2 ,⋅ ⋅⋅,g + ∆g,⋅ ⋅⋅, zq + ∆zq , λ( )[ ] (3.4)

    where the functional dependencies of R have been indicated explicitly.

    In principle, it should be possible to obtain ∆g by calculating the response

    for all possible thickness combinations weighted by the distribution functions

    representing the process-induced thickness variation of each layer. This approach

    is computationally impractical when q becomes large (for instance, if the cavity

    mirrors are made using multilayer dielectric mirrors).

    The uncertainty ∆g in gap can be analytically expressed using a Taylor

    series approximation

  • 26

    I R z1 + ∆z1, z2 + ∆z2 ,⋅ ⋅⋅,g + ∆g,⋅ ⋅⋅, zq + ∆zq , λ( )[ ] ≈ I R z1, z2 ,⋅ ⋅⋅,g,⋅ ⋅⋅, zq , λ( )[ ] + ∂ I∂ zi

      

      

    ⋅ ∆zi + i =1 i ≠ k

    q ∑ ∂ I∂ g

      

      

    ⋅ ∆g . (3.5)

    Combining equations (3.4) and (3.5) then gives ∆g;

    ∆g ≈ − ∂ I ∂ g

     

     

     

     

    −1 ⋅ ∆zi ⋅

    ∂ I ∂ zi

     

     i =1

    i ≠ k

    q

    ∑ . (3.6)

    To bound the uncertainty in gap, we should find the combination of layer

    thicknesses that maximizes ∆g for a given set of ∆zi; examination of equation

    (3.6) gives this bound, ∆gproc,

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